scholarly journals Pointed Hopf algebras: a guided tour to the liftings

2019 ◽  
Vol 53 (supl) ◽  
pp. 1-44 ◽  
Author(s):  
Iván Angiono ◽  
Agustín García Iglesias
Keyword(s):  
Hopf Algebras ◽  
The Other ◽  
Cartan Type ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
The One ◽  
Guided Tour

This article serves a two-fold purpose. On the one hand, it is asurvey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of Cartan type B2.

scholarly journals On the classification of finite-dimensional pointed Hopf algebras

2010 ◽  
Vol 171 (1) ◽  
pp. 375-417 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Hans-Jürgen Schneider
Keyword(s):  
Hopf Algebras ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras

Affine distance-transitive graphs with quadratic forms

1992 ◽  
Vol 112 (3) ◽  
pp. 507-517 ◽  
Author(s):  
John Van Bon
Keyword(s):  
Vector Space ◽  
Quadratic Forms ◽  
The Other ◽  
Almost Simple Group ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The One ◽  
Transitive Graphs

The classification of all finite primitive distance-transitive graphs is basically divided into two cases. In the one case, known as the almost simple case, we have an almost simple group acting primitively as a group of automorphisms on the graph. In the other case, known as the affine case, the vertices of the graph can be identified with the vectors of a finite-dimensional vector space over some finite field. In this case the automorphism group G of the graph Γ contains a normal p-subgroup N which is elementary Abelian and acts regularly on the set of vertices of Γ. Let G0 be the subgroup of G that stabilizes a vertex. Identifying the vertices of Γ with G0-cosets in G, one obtains a vector space V on which N acts as a group of translations, G0, stabilizes 0 and, as Γ is primitive, G0 acts irreducibly on V.

scholarly journals Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Chen Qian ◽  
Jing Wang
Keyword(s):  
Root System ◽  
Hopf Algebras ◽  
Enveloping Algebras ◽  
Finite Dimensional ◽  
Quantized Enveloping Algebras ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Simply Connected

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.

scholarly journals ON NICHOLS ALGEBRAS OVER PGL(2, q) AND PSL(2, q)

2010 ◽  
Vol 09 (02) ◽  
pp. 195-208 ◽  
Author(s):  
SEBASTIÁN FREYRE ◽  
MATÍAS GRAÑA ◽  
LEANDRO VENDRAMIN
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Drinfeld Modules ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Mathieu Groups ◽  
Nichols Algebras

We compute necessary conditions on Yetter–Drinfeld modules over the groups PGL(2, q) = PGL(2, 𝔽q) and PSL(2, q) = PSL(2, 𝔽q) to generate finite-dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that any finite-dimensional pointed Hopf algebra over the Mathieu groups M20 or M21 = PSL(3, 4) is the group algebra.

Somatoform disorders: diseases of the civilization

2020 ◽  
pp. 25-30
Author(s):  
I. Kukhtevich
Keyword(s):  
Health Professionals ◽  
Somatoform Disorders ◽  
Neuropsychiatric Disorders ◽  
The Other ◽  
Significant Part ◽  
Organic Basis ◽  
Autonomic Disorders ◽  
The One ◽  
Somatic Diseases

Functional autonomic disorders occupy a significant part in the practice of neurologists and professionals of other specialties as well. However, there is no generally accepted classification of such disorders. In this paper the authors tried to show that functional autonomic pathology corresponds to the concept of somatoform disorders combining syndromes manifested by visceral, borderline psychopathological, neurological symptoms that do not have an organic basis. The relevance of the problem of somatoform disorders is that on the one hand many health professionals are not familiar enough with manifestations of borderline neuropsychiatric disorders, often forming functional autonomic disorders, and on the other hand they overestimate somatoform symptoms that are similar to somatic diseases.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

2013 ◽  
Vol 12 (04) ◽  
pp. 1250191
Author(s):  
XIAOLAN YU ◽  
YINHUO ZHANG
Keyword(s):  
Spectral Sequence ◽  
Hopf Algebras ◽  
Type A ◽  
Serre Spectral Sequence ◽  
Nichols Algebra ◽  
Cartan Type ◽  
Dynkin Diagrams ◽  
Full Structure ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.

N. Belknap's Four-Valued Logic, Belknap Computer and New Proximity Functions for Comparing Discrete Objects

2021 ◽  
Vol 1 (2 (6)) ◽  
Author(s):  
Valerii Dmitrienko ◽  
Sergey Leonov ◽  
Mykola Mezentsev
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
The Other ◽  
Proximity Function ◽  
Input Information ◽  
Proximity Measures ◽  
The One ◽  
True Values

The idea of ​​Belknap's four-valued logic is that modern computers should function normally not only with the true values ​​of the input information, but also under the conditions of inconsistency and incompleteness of true failures. Belknap's logic introduces four true values: T (true - true), F (false - false), N (none - nobody, nothing, none), B (both - the two, not only the one but also the other).  For ease of work with these true values, the following designations are introduced: (1, 0, n, b). Belknap's logic can be used to obtain estimates of proximity measures for discrete objects, for which the functions Jaccard and Needhem, Russel and Rao, Sokal and Michener, Hamming, etc. are used. In this case, it becomes possible to assess the proximity, recognition and classification of objects in conditions of uncertainty when the true values ​​are taken from the set (1, 0, n, b). Based on the architecture of the Hamming neural network, neural networks have been developed that allow calculating the distances between objects described using true values ​​(1, 0, n, b). Keywords: four-valued Belknap logic, Belknap computer, proximity assessment, recognition and classification, proximity function, neural network.

scholarly journals Verbal Monomials in English for Audit and Accounting

2021 ◽  
Vol 04 (03) ◽  
Author(s):  
Oksana Chaika ◽  
Keyword(s):  
Functional Properties ◽  
Functional Property ◽  
The Other ◽  
Discussion Section ◽  
Work In Progress ◽  
The One

The paper research is work in progress and makes part of a publication set devoted the study of the English monomials and polynomials in the professional domain of audit and accounting, on the one hand. On the other, the research can be treated as a standalone piece for the study into the nature of verbal monomials as set term clusters in English for Audit and Accounting. The scope of research arrives at the following objectives. One objective is to give an overview of the term ‘monomial’ in English for Audit and Accounting, or English for A&A, which leads to understanding of the verbal monomial in English for A&A, correspondingly. The other objective refers to the classification introduced earlier as attributable to the analysis of the structure of the mentioned monomials and polynomials in English for A&A from a morphological perspective of the head term in a monomial, i.e. nounal, verbal, adjectival and adverbial. The said classification in this work associates with verbal monomials in English for A&A only, and provides a relevant sub-classification of the relevant verbal monomials through the lens of their functional properties and roles in a sentence, under the professional language framework. The results and discussion section presents five distinct groups of verbal monomials in English for Audit and Accounting, each corresponding to a specific syntactical role and functional property in a sentence. A variety of the examples helps see and identify the type of the English verbal monomial in the area of audit and accounting.

The Two Kinds of Depression According to St. Paul

1967 ◽  
Vol 113 (500) ◽  
pp. 779-780 ◽  
Author(s):  
Mark D. Altschule
Keyword(s):  
The Other ◽  
The Bible ◽  
The World ◽  
Current Classification ◽  
English Translations ◽  
The One ◽  
Similar Classification

One current classification of depression divides the syndrome into psychotic and non-psychotic varieties. It is interesting that a similar classification developed over a thousand years ago out of some words of St. Paul. In his Second Epistle to the Corinthians, Ch. 7, v. 10, Paul wrote: “For godly sorrow worketh repentance to salvation not to be repented of, but the sorrow of the world worketh death.” The word sorrow used in English translations of the Bible stood for the tristitia of Latin versions (Greek λνπη); connoting sadness, sorrow, despondency, depression. Paul's distinction between the two kinds of tristitia, the one “from God” and the other “of the world”, led mediaeval theologians to enlarge on differences between the two kinds of depression.

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